How can I increase the thickness of this object without breaking it?
I'm not 100% sure if this would work, but you might try it (quite possibly with the solidify modifier off, or perhaps only turned off after step 3.) and see if it does:


In Edit Mode, select all vertices of an object. 
Extrude, but don't move the extrusion. Simply click to finalize. This will create
duplicate vertices, which in this case, you actually want. Duplicating the selection might also work. 
Lastly, scale up. Only the duplicate vertices should be selected after the extrusion
operation, so scaling will move them out and create a "shell" of
sorts. 
Then add or remove geometry as needed to fill in gaps this
process might create.


Not gonna lie, if this works it's a bit of a messy solution. Make sure you have a backup save so that if you don't like the results you can revert back.